### Worms

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

### Greetings

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How many such students are there?

### Writ Large

Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

# Nice or Nasty for Two

##### Age 7 to 14 Challenge Level:

Here’s a game to play with an adult!

How do you play?
You'll need an adult to play with.
You’ll also need a $1$−$6$ dice, or even a $0$−$9$ dice if you have one. You could use a spinner instead of a dice. You could also use the interactive dice here.

Each of you draw a set of four boxes like this:

Or you could print off a scoring sheet.

Game $1$
Take turns with the adult to roll the dice and decide which of your four boxes to fill. Do this four times each until all your boxes are full. Read the four digits as a whole number.

Whoever has the larger four-digit number wins.

There are two possible scoring systems:
• A point for a win. The first person to reach $10$ wins the game
• Work out the difference between the two four-digit numbers after each round.
The winner keeps this score. First to $10000$ wins.

There are also some variations...

Game $2$
Whoever makes the smaller four digit number wins. See if you can come up with a suitable scoring system.

Game $3$
Set a target to aim for. Then throw the dice four times each and work out how far each of you is from the target number. Whoever is the closer wins.

There are two possible scoring systems:
• A point for a win. The first person to reach $10$ wins the game
• Work out the difference between the two four-digit numbers and the target number after each round. Keep a running total. First to $10000$ loses.

Game $4$
This game introduces a decimal point. The decimal point will take up one of the cells so this time the dice only needs to be thrown three times by each player. The winner is the one closer to the target. Choose a target.

Two possible versions:
• each player decides in advance where they want to put the decimal point before taking turns to throw the dice
• each player throws the dice three times and then decides where to place the digits and the decimal point.

Again, two different scoring systems are possible.

Game $5$
This is the nasty version!

Play any of the games above. This time you can choose to keep your number and put it in one of your cells, OR give it to the adult and tell them which cell to put it in! It's really important to take turns to start each round if this game is going to be fair.

What winning strategies can you come up with?
Why are some cells more significant than others?